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Solution of the Span-Wagner equation of state using

a density-energy state function for fluid-dynamic

simulation of carbon dioxide

Knut Erik Teigen Giljarhus, Svend Tollak Munkejord, and Geir Skaugen

SINTEF Energy Research, P.O. Box 4761 Sluppen, NO-7465 Trondheim, Norway

E-mail: Knut.Erik.Giljarhus@sintef.no; Svend.T.Munkejord@sintef.no; Geir.Skaugen@sintef.no

Abstract

With the introduction of carbon capture and storage (CCS) as a means to reduce carbon

emissions, a need has arisen for accurate and efficient simulation tools. In this work, we propose

a method for dynamic simulations of carbon dioxide using the SpanWagner reference equation

of state. The simulations are based on using the density and internal energy as states, which

is a formulation naturally resulting from mass and energy balances. The proposed numerical

method uses information about saturation lines to choose between single-phase and two-phase

equation systems, and is capable of handling phase transitions. To illustrate the potential of the

method, it is applied to simulations of tank depressurization, and to fluid-dynamic simulations

of pipe transport.

To whom correspondence should be addressed

1

Introduction

Carbon capture and storage (CCS) has been proposed as a strategy to reduce carbon emissions

to the atmosphere. An important part of the CCS chain is transportation, either in pipelines or in

tanks (on boats, vehicles or trains), from a capture point to a storage site. To ensure efficiency and

safety in these operations, it is important to have accurate simulation tools to control the processes,

and also to perform risk analysis. For instance, an issue with transportation of the supercritical

substance (CO2 is transported at high pressure, in order to minimize pipeline dimensions) is crack

propagation. If a pipeline ruptures, the crack may propagate along the pipe depending on the speed

of the crack relative to the speed of the expansion wave inside the pipe.1,2 To simulate this process,

an accurate model for the flow inside the pipe is needed. Another example is the global trade

of CO2 quotas, which has also been suggested as a means to reduce CO2 emissions.3 From an

economic perspective, it is important to accurately describe the CO2 properties in order to determine

the amount of CO2 stored and transported.

To perform a dynamic simulation, an appropriate mass balance and energy balance is often required

for the system of interest. As an example, consider the depressurization of a rigid tank as illustrated

in Figure 1. We can set up the following mass balance and energy balance for the problem:

Vddt

=m(T,), (1a)

dUdt

= Q(T ) m(T,)h(T,). (1b)

Here, V is the total volume of the tank, which is constant since the tank is assumed rigid, is

the density, m is the mass flow rate, U is the internal energy, Q is the heat transfer rate, T is

the temperature and h is the specific enthalpy. At each time step of a dynamic simulation, we

know and U . In order to update these, we need to find the temperature and enthalpy from an

equation of state (EOS). The fact that none of the intensive variables are known makes this problem

harder to solve than other formulations. The equilibrium condition represents a maximum in the

2

Q V

m

Figure 1: Schematic illustration of a tank.

entropy of the system.4 This problem has only been sporadically studied in the literature, and

usually for multi-component mixtures. In the context of multi-component mixtures, it is known

as the isoenergetic-isochoric flash, or the UV nnn flash, where nnn is the total number of moles of

each component. In this work, however, we work with systems of the type considered in Eq. (1),

which translates into a u problem, where u = U/(V ) is the specific internal energy. The first

appearance of the UV nnn-formulation known to the authors is Gani et al.,5 where it is argued that this

formulation has advantages over more traditional approaches. In Flatby et al.,6 dynamic simulations

of propane-butane mixtures in distillation columns were performed with a UV nnn-flash algorithm

based on a Newton-Raphson procedure. In Eckert and Kubcek,7 a rigorous model for simulating

multiple gas-liquid equilibrium stages was developed. This model includes complex multi-phase

systems of type gas-liquid-liquid and gas-liquid-liquid-liquid. New phases are dynamically removed

and added during the simulation. In Mller and Marquardt,8 a faster algorithm for the determination

of the number of phases was introduced. In Saha and Carroll,9 a partial Newton method was used,

along with a novel procedure for generating good initial estimates, to simulate the dynamic filling

of a vessel with nitrogen. In Michelsen,4 a more sophisticated flash algorithm is devised, which

uses stability analysis to determine the number of phases, and a combined solution procedure where

a nested-loop procedure is used together with the Newton method to give a robust method with

good performance. In Gonalves et al.,10 a similar algorithm was used to simulate storage tanks

and flash drums. In Castier,11 an optimization algorithm is used to directly maximize the entropy.

3

This algorithm is used in Castier 12 to simulate vessels with various fluid mixtures. In Arendsen

and Versteeg,13 the UV nnn formulation is used to simulate a liquefied gas tank with a propane-butane

mixture.

In this work, we are interested in the dynamic simulations involving pure CO2. This allows

several simplifications to be made compared to a multi-component mixture. Most importantly, the

composition is constant (since we only have one component), so saturation curves can be generated

a priori and this information can be utilized during simulation. To obtain the thermodynamic

properties, we use the SpanWagner equation of state. In Span and Wagner,14 an extensive body

of experimental data for CO2 was reviewed, and an EOS fitted to these data was presented. The

fitting procedure was based on the thermal properties of the single-phase region, the liquid-vapor

saturation curve, the speed of sound, the specific heat capacities, the specific internal energy and

the Joule-Thomson coefficient. The equation gives very accurate results, even in the region around

the critical point, and is now considered as the most accurate reference equation for CO2. It is

for instance used in the NIST Chemistry WebBook.15 There are other alternatives available that

are simpler computationally while retaining most of the accuracy, for instance Huang et al. 16 and

Kim,17 and the proposed method can be used with these equations of state as well.

In this work, we present an efficient method for solving the u problem for pure CO2 using the

SpanWagner (SW) EOS. Although the equation is complicated and contains many terms, we

demonstrate that it is feasible to use it in dynamic simulations. We compare the speed of the method

to an equivalent procedure from the NIST software package REFPROP.18 We also compare the

results to results obtained with simpler equations of state, the stiffened gas (SG) EOS19,20 and

the PengRobinson (PR) cubic EOS.21 The presented method can also easily be used with other

equations of state, for instance the two alternative equations of state mentioned above,16,17 or

equations of state for other substances like the reference equation of state for nitrogen from Span

et al..22 It can also be applied to multi-component mixtures, if the composition is constant. To

demonstrate the utility of the proposed method, we apply the algorithm to the dynamic simulation of

4

tank filling, and to a fluid-dynamic simulation of pipe transport. In a fluid-dynamic simulation, there

can be sharp gradients in the solution due to shock waves and other discontinuities. Additionally,

the simulation can potentially be time consuming due to a large number of cells and small time

steps. This puts high demands on the accuracy, robustness and efficiency of the methods used.

Thermodynamics

Thermodynamic properties

If an expression for the (specific) Helmholtz free energy, a(T,) is known, along with its derivatives,

all other thermodynamic properties can be derived from this expression. Important properties in this

work are pressure, specific entropy, specific enthalpy and specific internal energy, which expressed

in terms of the Helmholtz function and its derivatives are

P(T,) = 2(

a

)T, (2)

s(T,) =(

aT

)

(3)

h(T,) = a+(

a

)TT

(aT

), (4)

u(T,) = aT(

aT

). (5)

It is common to define a non-dimensional Helmholtz function, = a/(RT ), and split this function

into an ideal-gas part, 0, and a residual part, r,

(,) = 0(,)+r(,). (6)

This reduced Helmholtz function is expressed in terms of the reduced density, = /c and the

inverse reduced temperature, = Tc/T . c and Tc are the critical density and the critical temperature,

5

respectively. The critical properties of CO2 are summarized in Table 1. Expressed in terms of the

reduced Helmholtz energy, the pressure, specific entropy, specific enthalpy and specific internal

energy are

P(,)RT

= 1+r

, (7)

s(,)R

= , (8)

h(,)RT

= 1+

+r

, (9)

u(,)RT

=

. (10)

The SpanWagner equation of stat